SIMPLE EXPERIMENT TO VERIFY IF GRAVITATIONAL RED-SHIFTING IS TRUE
ABSTRACT
This home experiment could prove if gravitational shift exists, and verify Δf/f = gh/c².
Even when simple, and using off-the-shelf elements,
the use of a large amount of
seconds allows the possible test of red-shifting, if
the difference in the pulses count
increases constantly and regularly every Tᴺ = c²/(f₁gh) sec = 40,000 sec, being
f₁
the frequency of a simple MW cavity oscillator,
tuned at 10.00 Ghz. The readings
of two 50 bit counters are periodically collected
by a PC, which displays the value
of the counters and its difference. The experiment
is based on the continuous
phase accumulation of a 10 Ghz signal and its
receiving part, 22.2 m high, where
the gravitational effect should manifest as
being Δf/f = - gh/c² = -2.49E-15.
FREQUENCIES AND DIFFERENCES
IN COUNT OF PULSES.
Box 1 operates with a frequency f₁ = 10.00000 Ghz. The pulses
to be counted
continuously in N₁ are 0.1 nsec pulses.
Box 2 operates with a frequency f₂, received from the 10.00000 Ghz transmitted from
the ground level, 22.2 m down. f₂ = f₁ + Δf = f₁ - 0.000025 Hz.
This is because Δf/f = - gh/c², if Einstein gravitational shift prediction is correct.
Then, f₂ = 9,999,999,999.999975 Hz, and Δf/f₁ = (f₂ - f₁)/f₁ = -2.49E-15
If Einstein's gravitational shift prediction doesn't exists, f₂ = f₁ = 10.000000 Ghz.
The difference |Δf₁| = 0.000025 Hz creates a difference ΔN = N₁-N₂ = +1 count,
accumulative with a period 1/|Δf₁| = 40,000 seconds. This means about +2 counts/day
or +15/week.
SUBSYSTEMS
Besides two DirecTV antennae, which are fed directly
by the MW power amplifiers,
the following subsystems are used:
Box 1: Contains the 10.00000 Ghz cavity oscillator (f₁), the N₁ counter, the count
memory, updated every 0.4295 seconds with timer TG₁, the data count N₁ serializer
and the link with the PC, where the packetized reading of N₁. It also contains the
DC power feed for the Box 2, and two outputs (USB ports O₁ and O₂) to allow the
PC to receive the packets from both boxes. Each packet contains 64 bits of data
digital count plus a sequential number.
Box 2: Contains a 10.00000 Ghz pass band filter, a Ghz power amplifier, the N₂
counter, the count memory, updated with TG₂ every 0.4295 seconds, the data
count N₂ serializer and the link with the PC, to send the packetized count.
The cable used for packetized data transfer to the PC, through Box 1, also contains
the DC power feed.
PC: Contains the SW to compute N₁ and N₂ every TG = 0.4295 seconds. This signal
alert PIC1 and PIC2 to transmit data to two separate USB ports of the PC.
With N₁ and N₂ refreshed every 0.4295 sec, the PC SW can compute differences
indefinitely (hours, days, months).
SOFTWARE: Besides elementary binary arithmetic, to present ΔN = N₁ - N₂ data, the
software has to make corrections due to different
delay sources in the analog domain:
1) 74 nsec for the signal propagation delay between
Antenna 1 and Antenna 2.
2) 20 nsec (to be measured correctly) between the
input of the N₁ counter and
the input of the N₂ counter, due to the MW filter, delays in the RF
amplifiers, etc.
The total delay is constant, and has to be deducted
from ΔN = N₁ - N₂, as it represents
a difference of about 940 pulses between both
counters.
Counter reading times TG₁ and TG₂, and general timing TG = 0.4295 sec
TG₁ and TG₂ are derived from the signals S₁ and S₂ by dividing each one by 2³².
The division creates low frequency gating signals, used for periodic reading of
each counter N. The very small difference
femtosecond, which makes Tᴳ₁ = Tᴳ₂ = Tᴳ, to compute the time of readings.
f₁ = 10,000,000,000.0000000 Hz
T₁ = 0.1E-09 s
Tᴳ₁ = T₁ . 2³² = T₁ . 4,294,967,296 = 0.429496729600000 s
f₂ = f₁ + Δf = f₁ - 0.0000249 Hz = 9,999,999,999.9999751 Hz
T₂ = 1/(f₁ + Δf) = 1/f₁ (1 - 2.49E-15) s = T₁ (1 + 2.49E-15) s = 0.1E-09 s + 2.49E-25 s
Tᴳ₂ = T₂ . 2³² = T₂ . 4,294,967,296 = 0.429496729600000 s + 1.069446857E-15 s
ΔTᴳ = Tᴳ₂ - Tᴳ₁ = 1.069446857E-15 s (≈ 1 femtosecond)
For any practical purpose, Tᴳ₂ = Tᴳ₁, so Tᴳ = 0.429496729600000 s ≈ 0.4295 s is the
gating signal that is used to store the counts at Box 1 and Box 2, as the true real time clock.
EXAMPLE OF READINGS IN THE
PC SCREEN
This is an example of some readings every random multiples
of 20,000 TG (8,590 sec):
If gravitational shift exist, then every Tᴺ = c²/(f₁gh) = 40,000 sec, ΔN increments in 1.
Time (s) N1 N2 ΔN
0 0
0 0
8590 85900000000000 85900000000000 0
34360 343600000000000 343600000000000 0
42950 429500000000000 429499999999999 1
60130 601300000000000 601299999999999 1
77310 773100000000000 773099999999999 1
85900 859000000000000 858999999999998 2
103080 1030800000000000 1030799999999998 2
120260 1202600000000000 1202599999999997 3
154620 1546200000000000 1546199999999997 3
163210 1632100000000000 163299999999996 4
197570 1975700000000000 1975699999999996 4
206160 2061600000000000 2061599999999995 5
240520 2405200000000000 2405199999999995 5
249110 2491100000000000 249199999999994 6
257700 2577000000000000 2576999999999994 6
THE PC SOFTWARE.
The experiment can prove if gravitational shift in
frequency of a single
EM wave exist, due to the simplicity of the
method: counting accumulative
difference of pulses between two 50 bits
counter.
As the PC receives, every 0.4295 seconds, the
readings of N₁ and N₂, it
can compute indefinitely the difference ΔN = N₁ - N₂, even if the counters
overflow and restart from zero.
Each packet received in the PC every TG
seconds is
time-stamped by the
PC RTC clock, which is accurate enough for the
magnitudes of time involved.
The PC can store more than 400,000
readings per day indefinitely, which allows
computing at which value of Δf₁/f₁ (- gh/c² or other), the count increments
in one.
The structure of the packet that the PC reads is:
[PIC Number]
[Packet Number]
[50 bit counter reading]
This packet is stored in the PC with his RTC (Real Time Clock) value at the moment
of reception. So, what is stored in the memory, for further calculations, is the following
data structure:
[PIC Number]
[Packet Number]
[50 bit counter reading]
[PIC Continuous Time]
[PC RTC timestamp]
As both Packet Number word on each PIC is reset to zero under command of the
operator of the system, the TRUE CLOCK (same for both Boxes) is embedded into
each packet prior to its transmission, and is THE ONE that has to be considered
(for the purity of the experiment) when making averages, display data, etc.
This clocks (one per PIC) compute time as TP = 0.4295 x [Packet Number] seconds,
and last 16,384 packets.
Is the duty of the PC, with his timestamp on each packet received, to increase the
[PIC Continuous Time] field by 7,037 seconds when TP restart from zero.
In this way, the "experiment time" can be computed forever, in units of 0.4295 seconds,
showing this time in the field [PIC Continuous Time], which is stored along the rest of
the packet data in an ad-hoc database.
The only clock valid for the experiment is the one present in the field [PIC Continuous Time].
In this way, the PC can compute the elapsed time between integer increments of the
difference between counters, with an error lower than Tᴳ/Tᴺ = 42.95/40,000 % = 0.01%.
And, as Tᴺ = c²/(f₁gh) = 1/Δf, if happens, "Einstein's right" can be measured with this error.
SIMPLE
MATHEMATICS BEHIND THE EXPERIMENT
Einstein gravitational red-shifting of
electromagnetic waves is expressed, for
low heights, as:
Δf₁/f₁ = (f₂ - f₁)/f₁ = - gh/c²
f₂ = f₁ - f₁gh/c²
For f₁ = 10.000000 Ghz and h = 22.2 m, f₂ = 9,999,999,999.999975 Hz
The tiny difference of 0.000025 Hz
can be measured digitally, only if enough time
is allowed to pass, so one integer increase of
counts happens due to Δf.
With the frequency f₁ used, the time Tᴺ to produce a change ΔN = +1 is
Tᴺ = 1/|Δf₁| = 1/|(f₂ - f₁)| = c²/(f₁gh) = 40,000 seconds. In
one day, ΔN = +2
and, in one week, ΔN = +15. The difference
increases every Tᴺ seconds until
the experiment is stopped, or something breaks
down.
If this happens, then the existence of the
gravitational red-shifting can be
considered as true, and Einstein
was right. If no counts are accumulated
in ΔN, except random glitches of +1, then the
gravitational shift doesn't exist.
ANALOG COUNTERPART OF THE DIGITAL DIFFERENCE
OF COUNTS
In the analog world, the signal at the bottom can be represented by:
S₁(t) = A₁ sinΦ₁(t) = A₁ sin(ω₁t)
The signal received at the top, after filtered and amplified, can be
represented by
S₂(t) = A₂ sinΦ₂(t) = A₂ sin(ω₂t + θ) = A₂ sin[(ω₁ + Δω) t + θ]
S₂(t) = A₂ sin[ω₁ (1
- gh/c²) t + θ]
The constant θ represents a constant number of radians, due
to
delays originated in:
1) The time invested for the signal S₁ to travel 22.2 m, which is
0.0000000740 sec. This time comprehend 740 cycles of S₁,
representing a constant delay of 4,649.56 radians.
2) The natural delay of electric signals propagating through the
coax
cables, the power amplifiers, the reception filter and other minor
sources of delay. It's estimated as being a constant value of 20
nsec
end-end. Traduced in radians, it represents 200 x 2π radians, or
1,256.64 radians.
θ = 4,649.56 rad + 1,256.64 rad = 5,906.2 rad
The phases Φ₁(t) and Φ₂(t) are a linear function of t, and increase
continuously, without end, unless the source is turned off or
something breaks down.
The phase difference between S₁(t) and S₂(t), therefore, increase
constantly with time:
ΔΦ
= Φ₂(t) - Φ₁(t) = ω₁t - ω₁gh t/c² + θ - ω₁t = - ω₁gh t/c² + θ
ΔΦ = - 0.000156451 t + 5,906.2
Taken in absolute value, |ΔΦ| = |-
0.000156451 t + 5,906.2|.
To make more clear the analogy with the digital word, |ΔΦ| has
to be expressed in units of 2π radians, so the difference is now
expressed as:
K(t) =|ΔΦ|/2π = |-2.49E-05 t
+ 940| = |- t/40,160.7232 + 940|
It is evident that the analog phase difference K(t)
(discretized)
increments in ONE CYCLE every 40,160.7232 seconds,
indefinitely, as time passes.
But 40,160.7232 seconds is just 1/|Δf₁| = 1/|f₂ - f₁| = c²/(f₁gh),
which contains the Einstein's expression for gravitational shift
in
the small height h.
